Variational Inference with Implicit Approximate Inference Models (WIP Pt. 2)

Comments

In [1]:
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
In [2]:
import numpy as np
import keras.backend as K

import matplotlib.pyplot as plt
import seaborn as sns

from scipy.stats import logistic, multivariate_normal, norm
from scipy.special import expit

from keras.models import Model, Sequential
from keras.layers import Activation, Dense, Dot, Input
from keras.utils.vis_utils import model_to_dot

from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation

from IPython.display import SVG

Using TensorFlow backend.
/Users/tiao/.virtualenvs/implicit/lib/python3.6/site-packages/IPython/html.py:14: ShimWarning: The `IPython.html` package has been deprecated since IPython 4.0. You should import from `notebook` instead. `IPython.html.widgets` has moved to `ipywidgets`.
  "`IPython.html.widgets` has moved to `ipywidgets`.", ShimWarning)
In [3]:
plt.style.use('seaborn-notebook')
sns.set_context('notebook')
# display animation inline
plt.rc('animation', html='html5')
In [4]:
np.set_printoptions(precision=2,
                    edgeitems=3,
                    linewidth=80,
                    suppress=True)
In [5]:
K.tf.__version__
Out[5]:
'1.2.1'
In [6]:
LATENT_DIM = 2
NOISE_DIM = 3
BATCH_SIZE = 128
D_BATCH_SIZE = 128
G_BATCH_SIZE = 128
PRIOR_VARIANCE = 2.

Bayesian Logistic Regression (Synthetic Data)

In [86]:
w_min, w_max = -5, 5
In [87]:
w1, w2 = np.mgrid[w_min:w_max:300j, w_min:w_max:300j]
In [88]:
w_grid = np.dstack((w1, w2))
w_grid.shape
Out[88]:
(300, 300, 2)
In [89]:
prior = multivariate_normal(mean=np.zeros(LATENT_DIM), 
                            cov=PRIOR_VARIANCE)
In [90]:
log_prior = prior.logpdf(w_grid)
log_prior.shape
Out[90]:
(300, 300)
In [91]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, log_prior, cmap='magma')

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [92]:
x1 = np.array([ 1.5,  1.])
x2 = np.array([-1.5,  1.])
x3 = np.array([- .5, -1.])
In [93]:
X = np.vstack((x1, x2, x3))
X.shape
Out[93]:
(3, 2)
In [94]:
y1 = 1
y2 = 1
y3 = 0
In [95]:
y = np.stack((y1, y2, y3))
y.shape
Out[95]:
(3,)
In [96]:
def log_likelihood(w, x, y):
    # equiv. to negative binary cross entropy
    return np.log(expit(np.dot(w.T, x)*(-1)**(1-y)))
In [97]:
llhs = log_likelihood(w_grid.T, X.T, y)
llhs.shape
Out[97]:
(300, 300, 3)
In [98]:
fig, axes = plt.subplots(ncols=3, nrows=1, figsize=(6, 2))
fig.tight_layout()

for i, ax in enumerate(axes):
    
    ax.contourf(w1, w2, llhs[::,::,i], cmap=plt.cm.magma)

    ax.set_xlim(w_min, w_max)
    ax.set_ylim(w_min, w_max)
    
    ax.set_title('$p(y_{{{0}}} \mid x_{{{0}}}, w)$'.format(i+1))
    ax.set_xlabel('$w_1$')    
    
    if not i:
        ax.set_ylabel('$w_2$')

plt.show()
In [104]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, np.sum(llhs, axis=2), 
                cmap=plt.cm.magma)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [105]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, 
            np.exp(log_prior+np.sum(llhs, axis=2)), 
            cmap=plt.cm.magma)

ax.plot(*np.vstack((x1,x2,x3)).T, 'ro')

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()

Model Definitions

Density Ratio Estimator (Discriminator) Model

$T_{\psi}(x, z)$

Here we consider

$T_{\psi}(w)$

$T_{\psi} : \mathbb{R}^2 \to \mathbb{R}$

In [106]:
discriminator = Sequential(name='discriminator')
discriminator.add(Dense(10, input_dim=LATENT_DIM, activation='relu'))
discriminator.add(Dense(20, activation='relu'))
discriminator.add(Dense(1, activation=None, name='logit'))
discriminator.add(Activation('sigmoid'))
discriminator.compile(optimizer='adam',
                      loss='binary_crossentropy',
                      metrics=['binary_accuracy'])
In [107]:
ratio_estimator = Model(
    inputs=discriminator.inputs, 
    outputs=discriminator.get_layer(name='logit').output)
In [108]:
SVG(model_to_dot(discriminator, show_shapes=True)
    .create(prog='dot', format='svg'))
Out[108]:
G 4390601896 dense_6_input: InputLayerinput:output:(None, 2)(None, 2)4656746112 dense_6: Denseinput:output:(None, 2)(None, 10)4390601896->4656746112 4656743088 dense_7: Denseinput:output:(None, 10)(None, 20)4656746112->4656743088 4705728888 logit: Denseinput:output:(None, 20)(None, 1)4656743088->4705728888 4705708240 activation_2: Activationinput:output:(None, 1)(None, 1)4705728888->4705708240
In [109]:
w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

Initial density ratio, prior to any training

In [110]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [111]:
discriminator.evaluate(prior.rvs(size=5), np.zeros(5))

5/5 [==============================] - 0s
Out[111]:
[0.45842784643173218, 1.0]

Approximate Inference Model

$z_{\phi}(x, \epsilon)$

Here we only consider

$z_{\phi}(\epsilon)$

$z_{\phi}: \mathbb{R}^3 \to \mathbb{R}^2$

In [112]:
inference = Sequential()
inference.add(Dense(10, input_dim=NOISE_DIM, activation='relu'))
inference.add(Dense(20, activation='relu'))
inference.add(Dense(LATENT_DIM, activation=None))
inference.summary()

_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
dense_8 (Dense)              (None, 10)                40        
_________________________________________________________________
dense_9 (Dense)              (None, 20)                220       
_________________________________________________________________
dense_10 (Dense)             (None, 2)                 42        
=================================================================
Total params: 302
Trainable params: 302
Non-trainable params: 0
_________________________________________________________________

The variational parameters $\phi$ are the trainable weights of the approximate inference model

In [113]:
phi = inference.trainable_weights
phi
Out[113]:
[<tf.Variable 'dense_8/kernel:0' shape=(3, 10) dtype=float32_ref>,
 <tf.Variable 'dense_8/bias:0' shape=(10,) dtype=float32_ref>,
 <tf.Variable 'dense_9/kernel:0' shape=(10, 20) dtype=float32_ref>,
 <tf.Variable 'dense_9/bias:0' shape=(20,) dtype=float32_ref>,
 <tf.Variable 'dense_10/kernel:0' shape=(20, 2) dtype=float32_ref>,
 <tf.Variable 'dense_10/bias:0' shape=(2,) dtype=float32_ref>]
In [114]:
SVG(model_to_dot(inference, show_shapes=True)
    .create(prog='dot', format='svg'))
Out[114]:
G 4638359224 dense_8_input: InputLayerinput:output:(None, 3)(None, 3)4706841152 dense_8: Denseinput:output:(None, 3)(None, 10)4638359224->4706841152 4638452536 dense_9: Denseinput:output:(None, 10)(None, 20)4706841152->4638452536 4638609928 dense_10: Denseinput:output:(None, 20)(None, 2)4638452536->4638609928
In [115]:
eps = np.random.randn(BATCH_SIZE, NOISE_DIM)
In [116]:
w_posterior_samples = inference.predict(eps)
w_posterior_samples.shape
Out[116]:
(128, 2)
In [117]:
w_prior_samples = prior.rvs(size=BATCH_SIZE)
w_prior_samples.shape
Out[117]:
(128, 2)
In [120]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, 
            np.exp(log_prior+np.sum(llhs, axis=2)), 
            cmap=plt.cm.magma)

ax.scatter(*w_posterior_samples.T, alpha=.6)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [122]:
fig, ax = plt.subplots(figsize=(5, 5))

w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma, animate=True)

scatter_posterior = ax.scatter(*w_posterior_samples.T, alpha=.8)
scatter_prior = ax.scatter(*w_prior_samples.T, alpha=.8)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

props = dict(boxstyle='round', facecolor='w', alpha=0.5)

t = ax.text(0.05, 0.85, 'step: 0', 
            transform=ax.transAxes, bbox=props)

plt.show()
Discriminator pre-training
In [164]:
def prior_samples_gen(batch_size):

    while True:
        yield prior.rvs(size=batch_size)
In [165]:
def posterior_samples_gen(inference_model, batch_size):

    while True:
        eps = np.random.randn(batch_size, NOISE_DIM)
        yield inference_model.predict(eps)
In [172]:
def discriminator_data_gen(inference_model, batch_size):
    
    for samples_prior, samples_posterior in zip(prior_samples_gen(batch_size), 
                                                posterior_samples_gen(inference_model, batch_size)):
        inputs = np.vstack((samples_prior, samples_posterior))
        targets = np.hstack((np.zeros(batch_size), np.ones(batch_size)))
        yield inputs, targets
In [184]:
h = discriminator.fit_generator(generator=discriminator_data_gen(inference, 128), steps_per_epoch=32, epochs=2)

Epoch 1/2
32/32 [==============================] - 0s - loss: 0.1257 - binary_accuracy: 0.9559     
Epoch 2/2
32/32 [==============================] - 0s - loss: 0.1169 - binary_accuracy: 0.9598
In [185]:
h.history['loss'][-1]
Out[185]:
0.11688009253703058
In [167]:
metrics = discriminator.train_on_batch(D_input, D_labels)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-167-0dfc15023395> in <module>()
----> 1 metrics = discriminator.train_on_batch(D_input, D_labels)

NameError: name 'D_input' is not defined
In [36]:
def animate(step):

    ax.cla()

    w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
    w_grid_ratio = w_grid_ratio.reshape(300, 300)

    ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)

    info_dict = dict(zip(discriminator.metrics_names, metrics))
    info_dict['step'] = step

    props = dict(boxstyle='round', facecolor='w', alpha=0.5)

    t = ax.text(0.05, 0.85, 'step: 0', 
                transform=ax.transAxes, bbox=props)

    scatter_posterior = ax.scatter(*w_posterior_samples.T, alpha=.8)
    scatter_prior = ax.scatter(*w_prior_samples.T, alpha=.8)
    
    return ax
In [37]:
FuncAnimation(fig, animate, frames=50, 
              interval=200, # 5 fps
              blit=False)
Out[37]:
In [38]:
fig, ax = plt.subplots(figsize=(7, 7))

w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

cset = ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma, animate=True)

scatter_posterior = ax.scatter(*w_posterior_samples.T, alpha=.8)
scatter_prior = ax.scatter(*w_prior_samples.T, alpha=.8)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

props = dict(boxstyle='round', facecolor='w', alpha=0.5)

t = ax.text(0.05, 0.85, 'step: 0', 
            transform=ax.transAxes, bbox=props)

plt.show()
In [39]:
cset.collections
Out[39]:
<a list of 8 mcoll.PathCollection objects>
In [40]:
from matplotlib.collections import PatchCollection
In [ ]:
In [41]:
dir(cset.collections[0])
Out[41]:
['_A',
 '__class__',
 '__delattr__',
 '__dict__',
 '__dir__',
 '__doc__',
 '__eq__',
 '__format__',
 '__ge__',
 '__getattribute__',
 '__getstate__',
 '__gt__',
 '__hash__',
 '__init__',
 '__init_subclass__',
 '__le__',
 '__lt__',
 '__module__',
 '__ne__',
 '__new__',
 '__reduce__',
 '__reduce_ex__',
 '__repr__',
 '__setattr__',
 '__sizeof__',
 '__str__',
 '__subclasshook__',
 '__weakref__',
 '_agg_filter',
 '_alpha',
 '_animated',
 '_antialiaseds',
 '_axes',
 '_bcast_lwls',
 '_clipon',
 '_clippath',
 '_contains',
 '_edge_default',
 '_edgecolors',
 '_facecolors',
 '_factor',
 '_get_bool',
 '_get_value',
 '_gid',
 '_hatch',
 '_hatch_color',
 '_is_filled',
 '_is_stroked',
 '_label',
 '_linestyles',
 '_linewidths',
 '_mouseover',
 '_offset_position',
 '_offsets',
 '_oid',
 '_original_edgecolor',
 '_original_facecolor',
 '_path_effects',
 '_paths',
 '_picker',
 '_pickradius',
 '_prepare_points',
 '_prop_order',
 '_propobservers',
 '_rasterized',
 '_remove_method',
 '_set_edgecolor',
 '_set_facecolor',
 '_set_gc_clip',
 '_sizes',
 '_sketch',
 '_snap',
 '_stale',
 '_sticky_edges',
 '_transOffset',
 '_transform',
 '_transformSet',
 '_transforms',
 '_uniform_offsets',
 '_url',
 '_urls',
 '_us_linestyles',
 '_us_lw',
 '_visible',
 'add_callback',
 'add_checker',
 'aname',
 'autoscale',
 'autoscale_None',
 'axes',
 'callbacksSM',
 'changed',
 'check_update',
 'clipbox',
 'cmap',
 'colorbar',
 'contains',
 'convert_xunits',
 'convert_yunits',
 'draw',
 'eventson',
 'figure',
 'findobj',
 'format_cursor_data',
 'get_agg_filter',
 'get_alpha',
 'get_animated',
 'get_array',
 'get_axes',
 'get_children',
 'get_clim',
 'get_clip_box',
 'get_clip_on',
 'get_clip_path',
 'get_cmap',
 'get_contains',
 'get_cursor_data',
 'get_dashes',
 'get_datalim',
 'get_edgecolor',
 'get_edgecolors',
 'get_facecolor',
 'get_facecolors',
 'get_figure',
 'get_fill',
 'get_gid',
 'get_hatch',
 'get_label',
 'get_linestyle',
 'get_linestyles',
 'get_linewidth',
 'get_linewidths',
 'get_offset_position',
 'get_offset_transform',
 'get_offsets',
 'get_path_effects',
 'get_paths',
 'get_picker',
 'get_pickradius',
 'get_rasterized',
 'get_sizes',
 'get_sketch_params',
 'get_snap',
 'get_transform',
 'get_transformed_clip_path_and_affine',
 'get_transforms',
 'get_url',
 'get_urls',
 'get_visible',
 'get_window_extent',
 'get_zorder',
 'have_units',
 'hitlist',
 'is_figure_set',
 'is_transform_set',
 'mouseover',
 'norm',
 'pchanged',
 'pick',
 'pickable',
 'properties',
 'remove',
 'remove_callback',
 'set',
 'set_agg_filter',
 'set_alpha',
 'set_animated',
 'set_antialiased',
 'set_antialiaseds',
 'set_array',
 'set_axes',
 'set_clim',
 'set_clip_box',
 'set_clip_on',
 'set_clip_path',
 'set_cmap',
 'set_color',
 'set_contains',
 'set_dashes',
 'set_edgecolor',
 'set_edgecolors',
 'set_facecolor',
 'set_facecolors',
 'set_figure',
 'set_gid',
 'set_hatch',
 'set_label',
 'set_linestyle',
 'set_linestyles',
 'set_linewidth',
 'set_linewidths',
 'set_lw',
 'set_norm',
 'set_offset_position',
 'set_offsets',
 'set_path_effects',
 'set_paths',
 'set_picker',
 'set_pickradius',
 'set_rasterized',
 'set_sizes',
 'set_sketch_params',
 'set_snap',
 'set_transform',
 'set_url',
 'set_urls',
 'set_visible',
 'set_zorder',
 'stale',
 'stale_callback',
 'sticky_edges',
 'to_rgba',
 'update',
 'update_dict',
 'update_from',
 'update_scalarmappable',
 'zorder']
In [42]:
import matplotlib.patches as patches
In [43]:
from matplotlib.collections import PathCollection
In [44]:
fig, ax = plt.subplots(figsize=(7, 7))

ax.add_collection(cset.collections[4])

plt.show()

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
<ipython-input-44-d9b87f692bb4> in <module>()
      1 fig, ax = plt.subplots(figsize=(7, 7))
      2 
----> 3 ax.add_collection(cset.collections[4])
      4 
      5 plt.show()

~/.virtualenvs/implicit/lib/python3.6/site-packages/matplotlib/axes/_base.py in add_collection(self, collection, autolim)
   1754             collection.set_label('_collection%d' % len(self.collections))
   1755         self.collections.append(collection)
-> 1756         self._set_artist_props(collection)
   1757 
   1758         if collection.get_clip_path() is None:

~/.virtualenvs/implicit/lib/python3.6/site-packages/matplotlib/axes/_base.py in _set_artist_props(self, a)
    922     def _set_artist_props(self, a):
    923         """set the boilerplate props for artists added to axes"""
--> 924         a.set_figure(self.figure)
    925         if not a.is_transform_set():
    926             a.set_transform(self.transData)

~/.virtualenvs/implicit/lib/python3.6/site-packages/matplotlib/artist.py in set_figure(self, fig)
    648         # to more than one Axes
    649         if self.figure is not None:
--> 650             raise RuntimeError("Can not put single artist in "
    651                                "more than one figure")
    652         self.figure = fig

RuntimeError: Can not put single artist in more than one figure
In [134]:
fig, ax = plt.subplots(figsize=(7, 7))

cset = ax.contourf(np.linspace(-3, 3, 32), np.linspace(-3, 3, 32), np.random.randn(32, 32), cmap='magma')
scat = ax.scatter(*np.random.randn(2, 128), alpha=.8)

ax.set_xlim(-3, 3)
ax.set_ylim(-3, 3)

plt.show()
In [138]:
def animate(step):

    ax.cla()
    
    ax.contourf(np.linspace(-3, 3, 32), np.linspace(-3, 3, 32), np.random.randn(32, 32), cmap='magma')
    ax.scatter(*np.random.randn(2, 128), alpha=.8)

    ax.set_xlim(-3, 3)
    ax.set_ylim(-3, 3)
    
    return scat
In [139]:
FuncAnimation(fig, animate, frames=25, 
              interval=200) # 5 fps
Out[139]:
In [ ]:

Variational Inference with Implicit Approximate Inference Models (WIP)

Comments

In [69]:
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
In [70]:
import matplotlib.pyplot as plt
import seaborn as sns

import numpy as np
import keras.backend as K

from keras.models import Model, Sequential
from keras.layers import Activation, Dense, Dot, Input
from keras.utils.vis_utils import model_to_dot

from scipy.stats import logistic, multivariate_normal, norm
from scipy.special import expit
from mpl_toolkits.mplot3d import Axes3D

from IPython.display import SVG
In [71]:
plt.style.use('seaborn-notebook')
sns.set_context('notebook')
In [74]:
np.set_printoptions(precision=2,
                    edgeitems=3,
                    linewidth=80,
                    suppress=True)
In [75]:
K.tf.__version__
Out[75]:
'1.2.1'
In [76]:
LATENT_DIM = 2
NOISE_DIM = 3
BATCH_SIZE = 128
D_BATCH_SIZE = 128
G_BATCH_SIZE = 128
PRIOR_VARIANCE = 2.
In [77]:
w_min, w_max = -5, 5
In [78]:
w1, w2 = np.mgrid[w_min:w_max:300j, w_min:w_max:300j]
In [79]:
w_grid = np.dstack((w1, w2))
w_grid.shape
Out[79]:
(300, 300, 2)
In [80]:
prior = multivariate_normal(mean=np.zeros(LATENT_DIM), 
                            cov=PRIOR_VARIANCE)
In [81]:
log_prior = prior.logpdf(w_grid)
log_prior.shape
Out[81]:
(300, 300)
In [82]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, log_prior, cmap='magma')

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [83]:
x1 = np.array([ 1.5,  1.])
x2 = np.array([-1.5,  1.])
x3 = np.array([- .5, -1.])
In [84]:
X = np.vstack((x1, x2, x3))
X.shape
Out[84]:
(3, 2)
In [85]:
y1 = 1
y2 = 1
y3 = 0
In [86]:
y = np.stack((y1, y2, y3))
y.shape
Out[86]:
(3,)
In [87]:
def log_likelihood(w, x, y):
    # equiv. to negative binary cross entropy
    return np.log(expit(np.dot(w.T, x)*(-1)**(1-y)))
In [88]:
llhs = log_likelihood(w_grid.T, X.T, y)
llhs.shape
Out[88]:
(300, 300, 3)
In [89]:
fig, axes = plt.subplots(ncols=3, nrows=1, figsize=(6, 2))
fig.tight_layout()

for i, ax in enumerate(axes):
    
    ax.contourf(w1, w2, llhs[::,::,i], cmap=plt.cm.magma)

    ax.set_xlim(w_min, w_max)
    ax.set_ylim(w_min, w_max)
    
    ax.set_title('$p(y_{{{0}}} \mid x_{{{0}}}, w)$'.format(i+1))
    ax.set_xlabel('$w_1$')    
    
    if not i:
        ax.set_ylabel('$w_2$')

plt.show()
In [90]:
fig, ax = plt.subplots(figsize=(6, 5))

c = ax.contourf(w1, w2, np.sum(llhs, axis=2), 
                cmap=plt.cm.magma)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.colorbar(c)
plt.show()
In [91]:
fig, ax = plt.subplots(figsize=(6, 5))

c = ax.contourf(w1, w2, 
                np.exp(log_prior+np.sum(llhs, axis=2)), 
                cmap=plt.cm.magma)
ax.plot(*np.vstack((x1,x2,x3)).T, 'ro')

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.colorbar(c)
plt.show()

Model Definitions

Density Ratio Estimator (Discriminator) Model

$T_{\psi}(x, z)$

Here we consider

$T_{\psi}(w)$

$T_{\psi} : \mathbb{R}^2 \to \mathbb{R}$

In [92]:
discriminator = Sequential(name='discriminator')
discriminator.add(Dense(10, input_dim=LATENT_DIM, activation='relu'))
discriminator.add(Dense(20, activation='relu'))
discriminator.add(Dense(1, activation=None, name='logit'))
discriminator.add(Activation('sigmoid'))
discriminator.compile(optimizer='adam',
                      loss='binary_crossentropy')
In [93]:
ratio_estimator = Model(
    inputs=discriminator.inputs, 
    outputs=discriminator.get_layer(name='logit').output)
In [94]:
SVG(model_to_dot(discriminator, show_shapes=True)
    .create(prog='dot', format='svg'))
Out[94]:
G 140717329860592 dense_6_input: InputLayerinput:output:(None, 2)(None, 2)140717330379888 dense_6: Denseinput:output:(None, 2)(None, 10)140717329860592->140717330379888 140717330379944 dense_7: Denseinput:output:(None, 10)(None, 20)140717330379888->140717330379944 140717329859024 logit: Denseinput:output:(None, 20)(None, 1)140717330379944->140717329859024 140717329765656 activation_2: Activationinput:output:(None, 1)(None, 1)140717329859024->140717329765656
In [95]:
w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

Initial density ratio, prior to any training

In [96]:
fig, ax = plt.subplots(figsize=(5, 5))

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [97]:
discriminator.evaluate(prior.rvs(size=5), np.ones(5))

5/5 [==============================] - 0s
Out[97]:
0.60724371671676636

Approximate Inference Model

$z_{\phi}(x, \epsilon)$

Here we only consider

$z_{\phi}(\epsilon)$

$z_{\phi}: \mathbb{R}^3 \to \mathbb{R}^2$

In [98]:
inference = Sequential()
inference.add(Dense(10, input_dim=NOISE_DIM, activation='relu'))
inference.add(Dense(20, activation='relu'))
inference.add(Dense(LATENT_DIM, activation=None))
inference.summary()

_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
dense_8 (Dense)              (None, 10)                40        
_________________________________________________________________
dense_9 (Dense)              (None, 20)                220       
_________________________________________________________________
dense_10 (Dense)             (None, 2)                 42        
=================================================================
Total params: 302
Trainable params: 302
Non-trainable params: 0
_________________________________________________________________

The variational parameters $\phi$ are the trainable weights of the approximate inference model

In [99]:
phi = inference.trainable_weights
phi
Out[99]:
[<tf.Variable 'dense_8/kernel:0' shape=(3, 10) dtype=float32_ref>,
 <tf.Variable 'dense_8/bias:0' shape=(10,) dtype=float32_ref>,
 <tf.Variable 'dense_9/kernel:0' shape=(10, 20) dtype=float32_ref>,
 <tf.Variable 'dense_9/bias:0' shape=(20,) dtype=float32_ref>,
 <tf.Variable 'dense_10/kernel:0' shape=(20, 2) dtype=float32_ref>,
 <tf.Variable 'dense_10/bias:0' shape=(2,) dtype=float32_ref>]
In [100]:
SVG(model_to_dot(inference, show_shapes=True)
    .create(prog='dot', format='svg'))
Out[100]:
G 140717330378936 dense_8_input: InputLayerinput:output:(None, 3)(None, 3)140717330676480 dense_8: Denseinput:output:(None, 3)(None, 10)140717330378936->140717330676480 140717329573760 dense_9: Denseinput:output:(None, 10)(None, 20)140717330676480->140717329573760 140718031384192 dense_10: Denseinput:output:(None, 20)(None, 2)140717329573760->140718031384192
In [101]:
eps = np.random.randn(BATCH_SIZE, NOISE_DIM)
In [102]:
w_posterior_samples = inference.predict(eps)
w_posterior_samples.shape
Out[102]:
(128, 2)
In [103]:
w_prior_samples = prior.rvs(size=BATCH_SIZE)
w_prior_samples.shape
Out[103]:
(128, 2)
In [104]:
fig, ax = plt.subplots(figsize=(6, 5))

c = ax.contourf(w1, w2, 
                np.exp(log_prior+np.sum(llhs, axis=2)), 
                cmap=plt.cm.magma)

ax.scatter(*w_posterior_samples.T, alpha=.6)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.colorbar(c)
plt.show()
In [105]:
fig, ax = plt.subplots(figsize=(5, 5))

w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)
ax.scatter(*w_posterior_samples.T, alpha=.6)
ax.scatter(*w_prior_samples.T, alpha=.6)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [106]:
sess = K.get_session()
In [107]:
w_prior_samples = K.random_normal(shape=(BATCH_SIZE, LATENT_DIM), 
                                  stddev=np.sqrt(PRIOR_VARIANCE))
In [108]:
eps = K.random_normal(shape=(BATCH_SIZE, NOISE_DIM))
In [109]:
w_posterior_samples = inference(eps)
w_posterior_samples
Out[109]:
<tf.Tensor 'sequential_2/dense_10/BiasAdd:0' shape=(128, 2) dtype=float32>
In [110]:
fig, ax = plt.subplots(figsize=(5, 5))

w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)
ax.scatter(*sess.run(w_posterior_samples).T, alpha=.6)
ax.scatter(*sess.run(w_prior_samples).T, alpha=.6)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [111]:
discrim_loss = K.mean(
    K.binary_crossentropy(
        discriminator(w_posterior_samples), 
        K.ones_like(discriminator(w_posterior_samples))) +
    K.binary_crossentropy(
        discriminator(w_prior_samples), 
        K.zeros_like(discriminator(w_prior_samples))))
In [112]:
discrim_loss.eval(session=sess)
Out[112]:
1.385956
In [113]:
opt = K.tf.train.AdamOptimizer(3e-3, beta1=0.9)
In [114]:
discrim_train_op = opt.minimize(discrim_loss, 
                                var_list=discriminator.trainable_weights)
In [115]:
K.mean(ratio_estimator(w_posterior_samples))
Out[115]:
<tf.Tensor 'Mean_14:0' shape=() dtype=float32>
In [116]:
K.expand_dims(K.constant(y), 1)
Out[116]:
<tf.Tensor 'ExpandDims_8:0' shape=(3, 1) dtype=float32>
In [117]:
K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1))
Out[117]:
<tf.Tensor 'Pow_7:0' shape=(3, 1) dtype=float32>
In [118]:
K.dot(K.constant(X), K.transpose(w_posterior_samples))
Out[118]:
<tf.Tensor 'MatMul_7:0' shape=(3, 128) dtype=float32>
In [119]:
K.dot(K.constant(X), K.transpose(w_posterior_samples))*K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1))
Out[119]:
<tf.Tensor 'mul_10:0' shape=(3, 128) dtype=float32>
In [120]:
K.sigmoid(K.dot(K.constant(X), K.transpose(w_posterior_samples)) *
          K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1)))
Out[120]:
<tf.Tensor 'Sigmoid_5:0' shape=(3, 128) dtype=float32>
In [121]:
K.log(K.sigmoid(K.dot(K.constant(X), K.transpose(w_posterior_samples)) *
                K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1))))
Out[121]:
<tf.Tensor 'Log_10:0' shape=(3, 128) dtype=float32>
In [122]:
K.mean(K.log(K.sigmoid(K.dot(K.constant(X), K.transpose(w_posterior_samples)) *
                       K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1)))))
Out[122]:
<tf.Tensor 'Mean_15:0' shape=() dtype=float32>
In [123]:
log_likelihood = K.mean(K.log(K.sigmoid(K.dot(K.constant(X), K.transpose(w_posterior_samples)) *
                                        K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1)))))
log_likelihood
Out[123]:
<tf.Tensor 'Mean_16:0' shape=() dtype=float32>
In [124]:
inference_loss = K.mean(ratio_estimator(w_posterior_samples)) - log_likelihood
In [125]:
inference_loss.eval(session=sess)
Out[125]:
0.62940556
In [126]:
inference_train_op = opt.minimize(inference_loss, 
                                  var_list=inference.trainable_weights)
In [127]:
keras_llh = K.reshape(K.sum(K.log(K.sigmoid(K.dot(K.constant(X), K.reshape(K.permute_dimensions(K.constant(w_grid), (2, 0, 1)), shape=(2, 300*300))) *
                                            K.pow(K.constant(-1), 1-K.expand_dims(K.constant(y), 1)))), 
                            axis=0), 
                      shape=(300, 300))
keras_llh
Out[127]:
<tf.Tensor 'Reshape_3:0' shape=(300, 300) dtype=float32>
In [128]:
fig, ax = plt.subplots(figsize=(6, 5))

c = ax.contourf(w1, w2, sess.run(keras_llh), cmap=plt.cm.magma)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.colorbar(c)
plt.show()
In [129]:
np.allclose(sess.run(keras_llh), 
            np.sum(llhs, axis=2))
Out[129]:
True
In [130]:
sess = K.get_session()
In [131]:
for d_step in range(3*300):
    loss, _ = sess.run([discrim_loss, discrim_train_op])
    print(loss)

1.38532
1.3579
1.37045
1.32634
1.33504
1.27227
1.28475
1.24287
1.22682
1.22881
1.21535
1.19214
1.15849
1.14416
1.14371
1.14085
1.09653
1.1075
1.10195
1.09052
1.05989
1.08887
1.04995
1.05183
1.05929
1.03833
1.01648
1.03497
1.01955
0.960189
0.988754
0.951096
1.00077
0.951274
0.980649
0.966437
0.972831
0.949386
0.957259
0.926495
0.937377
0.916226
0.937085
0.914591
0.889111
0.904115
0.874599
0.862437
0.820481
0.895674
0.840388
0.906279
0.875646
0.839755
0.848321
0.792723
0.82651
0.818164
0.811942
0.773857
0.748018
0.803157
0.784107
0.763402
0.749615
0.775418
0.760771
0.789065
0.773577
0.716524
0.741916
0.747315
0.7378
0.732429
0.715211
0.701757
0.66048
0.696592
0.67441
0.715436
0.752388
0.672804
0.655467
0.693615
0.662632
0.660639
0.670306
0.625209
0.672787
0.661013
0.612875
0.636962
0.598916
0.662136
0.55278
0.613553
0.58413
0.567281
0.625276
0.573121
0.602104
0.579841
0.568854
0.565051
0.569442
0.54241
0.546366
0.538198
0.552248
0.601788
0.509559
0.536433
0.548869
0.538587
0.507484
0.508999
0.503613
0.476276
0.499506
0.508605
0.467023
0.461849
0.481012
0.472659
0.460581
0.45003
0.499749
0.454548
0.449862
0.536963
0.448475
0.467067
0.448574
0.486148
0.452766
0.437414
0.456874
0.459249
0.422732
0.420728
0.440434
0.461794
0.42356
0.463026
0.333407
0.433452
0.452114
0.35519
0.38314
0.367164
0.404151
0.342661
0.333128
0.346002
0.371624
0.452988
0.459696
0.428007
0.354236
0.311349
0.367938
0.462415
0.324165
0.452471
0.370054
0.333621
0.340891
0.447923
0.442891
0.336959
0.304018
0.333281
0.329424
0.321482
0.336908
0.375099
0.281343
0.3104
0.393116
0.393949
0.254872
0.444985
0.397345
0.354726
0.345908
0.228596
0.398022
0.353818
0.270504
0.336221
0.404845
0.321817
0.377144
0.444702
0.349187
0.339988
0.323128
0.402443
0.300859
0.288087
0.25431
0.299444
0.255488
0.313054
0.389661
0.230185
0.415969
0.392863
0.321102
0.322946
0.318168
0.248868
0.430951
0.271098
0.340435
0.342456
0.25684
0.340246
0.372529
0.45213
0.223858
0.288524
0.313052
0.232402
0.330251
0.378014
0.263161
0.252301
0.291386
0.300431
0.262314
0.434332
0.367387
0.252038
0.244233
0.402506
0.345745
0.255478
0.262207
0.311652
0.336827
0.268766
0.364561
0.325013
0.393565
0.361726
0.357331
0.233737
0.320791
0.228293
0.282431
0.288983
0.407007
0.337941
0.304532
0.271827
0.267347
0.362266
0.304938
0.265404
0.255674
0.355238
0.406194
0.256441
0.314476
0.290617
0.361067
0.230431
0.231442
0.214638
0.27529
0.276817
0.288087
0.33216
0.302738
0.240562
0.342358
0.321828
0.431387
0.250847
0.27855
0.244946
0.269892
0.392373
0.289954
0.280752
0.379365
0.308509
0.380273
0.2464
0.327529
0.320829
0.293981
0.174126
0.365427
0.220253
0.333885
0.271242
0.308914
0.276282
0.266287
0.311934
0.348004
0.275128
0.207578
0.413157
0.224726
0.226337
0.253312
0.241829
0.315702
0.214764
0.326352
0.386239
0.291657
0.272628
0.346335
0.245491
0.334242
0.258078
0.354176
0.321975
0.258085
0.295802
0.305626
0.387766
0.273724
0.290287
0.321441
0.327561
0.255026
0.290386
0.280339
0.255527
0.298302
0.306683
0.195443
0.358449
0.241336
0.373126
0.253557
0.214792
0.305674
0.280378
0.343197
0.366787
0.276942
0.268094
0.168371
0.245901
0.298778
0.259956
0.25666
0.287455
0.269845
0.218483
0.294437
0.295181
0.305536
0.206319
0.280013
0.322677
0.288346
0.241131
0.266195
0.264982
0.296257
0.3003
0.377406
0.26884
0.268009
0.287718
0.227764
0.327918
0.207449
0.336152
0.234233
0.30972
0.37544
0.203731
0.294301
0.313933
0.271753
0.389465
0.412929
0.304692
0.298007
0.343537
0.255937
0.154347
0.317867
0.249846
0.345872
0.246336
0.277842
0.240931
0.298713
0.311668
0.317681
0.308057
0.256132
0.230962
0.365042
0.326694
0.182882
0.298977
0.230151
0.319488
0.302716
0.337055
0.280472
0.330269
0.271947
0.253045
0.201525
0.212759
0.329366
0.2667
0.215311
0.318515
0.307372
0.270939
0.322351
0.326016
0.224039
0.200143
0.148069
0.322914
0.13252
0.305279
0.34209
0.192442
0.234607
0.224739
0.314081
0.216559
0.313695
0.287918
0.219836
0.280369
0.21938
0.330277
0.298507
0.229414
0.364292
0.321173
0.315297
0.247579
0.200411
0.318642
0.303608
0.204451
0.199351
0.328443
0.221468
0.234873
0.31292
0.257741
0.246582
0.175981
0.323016
0.331048
0.308028
0.357653
0.174243
0.24941
0.22787
0.237612
0.274157
0.266045
0.278884
0.30807
0.224181
0.21667
0.219935
0.215113
0.295749
0.228779
0.287847
0.274755
0.290104
0.257592
0.158541
0.307881
0.25223
0.246907
0.207552
0.256707
0.271685
0.263728
0.195208
0.297048
0.323941
0.256202
0.296758
0.304664
0.282952
0.325998
0.222861
0.298897
0.260353
0.22891
0.236029
0.267145
0.257044
0.235449
0.190918
0.327772
0.339791
0.271775
0.258712
0.315136
0.292156
0.223943
0.253757
0.249119
0.329328
0.240682
0.239011
0.330561
0.250688
0.275255
0.194366
0.217429
0.278699
0.363364
0.38697
0.241689
0.220995
0.258278
0.245144
0.284179
0.229927
0.172926
0.252574
0.335411
0.254785
0.348012
0.214253
0.278575
0.238369
0.257474
0.271734
0.243277
0.277504
0.341744
0.319304
0.317398
0.280298
0.340003
0.28844
0.218135
0.293611
0.336241
0.241568
0.354696
0.218075
0.284119
0.256733
0.215865
0.418464
0.153256
0.277758
0.34079
0.268386
0.207642
0.248794
0.214411
0.29601
0.309726
0.199095
0.320172
0.192783
0.197918
0.377143
0.2568
0.177411
0.351495
0.130222
0.340184
0.291682
0.290544
0.227707
0.278339
0.292904
0.369405
0.232014
0.243364
0.261748
0.273672
0.213081
0.235931
0.181272
0.289088
0.276575
0.174001
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0.187932
0.282802
0.239955
0.284414
0.243658
0.221988
0.258108
0.246219
0.289428
0.257696
0.229551
0.21741
0.328822
0.311971
0.254872
0.316906
0.231207
0.262648
0.306369
0.288493
0.271459
0.192031
0.303622
0.41162
0.223178
0.259723
0.214679
0.397215
0.174605
0.295229
0.268848
0.228744
0.264676
0.265323
0.2117
0.24944
0.200965
0.338666
0.355019
0.345025
0.30573
0.21673
0.330682
0.274113
0.192523
0.270616
0.289474
0.211751
0.252088
0.231713
0.332725
0.199363
0.250235
0.288874
0.22122
0.276812
0.265646
0.223562
0.262645
0.350757
0.265914
0.22279
0.334539
0.28883
0.286489
0.25148
0.410521
0.221324
0.322639
0.262011
0.209446
0.189491
0.23558
0.220028
0.192532
0.163899
0.271476
0.24814
0.338593
0.256927
0.251407
0.267802
0.290043
0.245387
0.262148
0.215823
0.261929
0.282015
0.211442
0.221292
0.340921
0.248831
0.277278
0.221039
0.216445
0.386351
0.212501
0.197268
0.285732
0.250364
0.285536
0.23757
0.269246
0.22951
0.183171
0.250872
0.30216
0.236106
0.358883
0.251053
0.266955
0.258212
0.367088
0.332073
0.323386
0.242664
0.226073
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0.304846
0.375678
0.330368
0.313987
0.19607
0.134187
0.268039
0.19912
0.301802
0.175763
0.146937
0.250498
0.286757
0.340791
0.19479
0.220702
0.417666
0.29547
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0.305307
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0.329802
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0.199395
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0.235254
0.276254
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In [132]:
fig, ax = plt.subplots(figsize=(5, 5))

w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)
ax.scatter(*sess.run(w_posterior_samples).T, alpha=.6)
ax.scatter(*sess.run(w_prior_samples).T, alpha=.6)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [133]:
for step in range(3*200):

    g_loss, _ = sess.run([inference_loss, inference_train_op])

    for d_step in range(3*50):

        d_loss, _ = sess.run([discrim_loss, discrim_train_op])
    
    print(d_loss, g_loss)

0.299976 3.34463
0.42477 3.42557
0.489073 3.16171
0.492646 2.78832
0.697428 2.15677
0.733206 1.72106
0.733298 1.52407
0.807746 1.29283
1.01707 1.36824
0.982902 1.16122
0.972628 1.10115
1.03581 1.21175
1.08277 1.57636
1.06537 1.18987
1.08656 1.15926
1.12824 1.12963
1.11872 1.36252
1.20765 1.06292
1.25455 1.00747
1.29707 1.10839
1.29748 1.12028
1.30085 0.914041
1.32332 0.986965
1.29178 0.985552
1.32019 1.00764
1.2936 1.06811
1.31705 1.0777
1.34883 1.0747
1.25078 0.984631
1.27866 1.05978
1.26078 1.02381
1.23433 0.950675
1.17123 0.905802
1.20704 1.02897
1.17519 0.958528
1.18514 0.944483
1.16954 0.834288
1.2178 0.968593
1.16209 0.839961
1.20248 0.932302
1.14983 0.903876
1.22635 0.947336
1.14538 1.02064
1.18963 0.978661
1.19453 0.916686
1.30689 0.953143
1.23519 0.983169
1.28594 0.952351
1.26316 0.985327
1.25242 0.95212
1.25774 0.928886
1.26975 0.893379
1.27822 0.898999
1.28621 0.888295
1.33779 0.949368
1.22241 0.955898
1.33705 0.829938
1.27576 0.985328
1.32099 0.844981
1.31612 0.929675
1.27131 0.906424
1.31378 0.931417
1.24642 0.892234
1.30976 0.890492
1.25219 0.913798
1.28763 0.873917
1.29087 0.897982
1.30312 0.900786
1.2624 0.86226
1.29503 0.897659
1.34493 0.907091
1.20294 0.901368
1.29374 0.888826
1.27329 0.933204
1.2393 0.855003
1.26956 0.798132
1.24166 0.98339
1.33536 0.908256
1.28197 0.909913
1.26196 0.928738
1.25027 0.899723
1.29102 0.904741
1.2854 0.895078
1.28242 0.932583
1.29949 0.93923
1.33095 0.933411
1.31844 0.906776
1.23557 0.914326
1.29453 0.88582
1.3628 0.929388
1.29 0.983679
1.28356 0.910055
1.29529 0.888072
1.27999 0.870004
1.2683 0.93354
1.28719 0.855926
1.25781 0.97036
1.23279 0.88822
1.29746 0.913348
1.1776 0.91485
1.23394 0.862434
1.24866 0.926131
1.26843 0.904428
1.28506 0.90711
1.28996 0.919347
1.30133 0.936005
1.19388 0.895903
1.29447 0.848564
1.29656 0.866625
1.30646 0.876477
1.34333 0.915293
1.31501 0.877347
1.29116 0.859644
1.24887 0.882841
1.24631 0.885617
1.29015 0.925848
1.31109 0.83671
1.27925 0.923947
1.1967 0.858732
1.27543 0.923373
1.30359 0.922775
1.32006 0.861679
1.28861 0.888932
1.25018 0.960168
1.28214 0.921384
1.2733 0.898344
1.24917 0.900367
1.29188 0.848968
1.2743 0.8913
1.2666 0.882793
1.31015 0.882384
1.33708 0.880494
1.33403 0.938472
1.30036 0.907784
1.2895 0.927621
1.22617 0.863041
1.2021 0.889226
1.29513 0.906224
1.21781 0.956144
1.24994 0.862738
1.28891 0.918831
1.26892 0.928247
1.29732 0.844343
1.28297 0.850404
1.31501 0.943559
1.2537 0.913817
1.22275 0.911185
1.31151 0.952311
1.27463 0.908037
1.30624 0.904855
1.24646 0.885781
1.28867 0.914277
1.2849 0.840575
1.35278 0.877746
1.2319 0.902562
1.31323 0.889612
1.26586 0.925208
1.32814 0.939342
1.28865 0.910893
1.3173 0.886235
1.26333 0.865478
1.23717 0.871529
1.23409 0.912993
1.30309 0.883133
1.28162 0.910106
1.28846 0.863426
1.30377 0.886678
1.23869 0.881498
1.27749 0.914509
1.35021 0.887712
1.27151 0.858169
1.24585 0.906618
1.28383 0.90202
1.29657 0.860148
1.3807 0.929281
1.23629 0.867771
1.26817 0.904284
1.27373 0.864512
1.23917 0.927002
1.27068 0.849798
1.31069 0.869198
1.35202 0.943909
1.22629 0.941014
1.23198 0.916758
1.31284 0.904447
1.25665 0.888998
1.24618 0.903868
1.35746 0.883824
1.26001 0.897694
1.26402 0.917581
1.25175 0.873426
1.26214 0.896461
1.24735 0.927409
1.25836 0.897048
1.32878 0.865081
1.25151 0.932814
1.31282 0.862772
1.29125 0.91402
1.29157 0.902586
1.3243 0.909463
1.26532 0.932931
1.25602 0.898872
1.31726 0.904133
1.34997 0.858721
1.32501 0.921767
1.30467 0.876867
1.26928 0.900812
1.27455 0.934993
1.26909 0.892616
1.29086 0.923215
1.3279 0.877819
1.22691 0.930301
1.32907 0.901793
1.27293 0.907734
1.30581 0.918265
1.26162 0.919997
1.28483 0.867349
1.28488 0.883433
1.28077 0.870668
1.18876 0.903459
1.25426 0.862026
1.26798 0.928929
1.31472 0.854127
1.27993 0.87624
1.27464 0.867176
1.29654 0.882978
1.32485 0.886917
1.26158 0.855248
1.24716 0.862936
1.28117 0.931524
1.20632 0.885412
1.20615 0.882369
1.23783 0.876454
1.30756 0.904714
1.26956 0.883896
1.24094 0.843747
1.23795 0.893069
1.28202 0.874765
1.23506 0.918747
1.22337 0.83743
1.3601 0.871304
1.24585 0.913947
1.24939 0.869258
1.31665 0.911868
1.29636 0.910884
1.2456 0.8817
1.27877 0.869583
1.22918 0.859839
1.26446 0.887992
1.19153 0.922629
1.27731 0.885781
1.24352 0.887125
1.27908 0.859451
1.29932 0.897275
1.29007 0.875759
1.31984 0.866664
1.27071 0.921769
1.22873 0.906817
1.28178 0.893999
1.21822 0.902762
1.27645 0.882699
1.33552 0.942864
1.29019 0.888469
1.26938 0.886925
1.33267 0.913781
1.23469 0.91071
1.24169 0.893096
1.27127 0.883584
1.27609 0.910159
1.24287 0.899395
1.27423 0.86692
1.26439 0.881307
1.27183 0.913777
1.32222 0.900624
1.30228 0.887633
1.3534 0.900266
1.28201 0.926251
1.24125 0.885375
1.27778 0.876981
1.34469 0.882942
1.20779 0.894412
1.29135 0.91812
1.36108 0.915039
1.25821 0.892367
1.27303 0.894492
1.27273 0.918805
1.35934 0.885502
1.22958 0.895696
1.31604 0.914404
1.30014 0.929029
1.32877 0.914306
1.2374 0.896187
1.29922 0.923526
1.31934 0.897103
1.25371 0.876657
1.27429 0.917426
1.19264 0.870283
1.21791 0.897719
1.28827 0.932747
1.29979 0.929157
1.34422 0.905238
1.27498 0.868193
1.32279 0.860391
1.30463 0.896259
1.28341 0.890942
1.26142 0.901491
1.29638 0.877999
1.2651 0.902819
1.31532 0.891434
1.26534 0.876294
1.27595 0.91941
1.29804 0.895347
1.35388 0.915468
1.30606 0.875761
1.2592 0.897856
1.28771 0.890425
1.25574 0.868415
1.30032 0.880557
1.24677 0.858536
1.32066 0.903354
1.34579 0.91493
1.21861 0.87927
1.27466 0.913171
1.26265 0.902121
1.30983 0.886372
1.25601 0.891181
1.21984 0.966641
1.27002 0.888857
1.33133 0.892649
1.2784 0.876429
1.34413 0.91359
1.28193 0.873876
1.33145 0.912212
1.33636 0.882785
1.18631 0.890936
1.29176 0.884748
1.30906 0.92735
1.27363 0.912045
1.29613 0.901083
1.27562 0.916263
1.2322 0.904348
1.33516 0.925911
1.28743 0.900918
1.2766 0.883029
1.34278 0.876535
1.20862 0.903666
1.25352 0.896885
1.2914 0.885103
1.22056 0.905637
1.27943 0.886183
1.29659 0.85713
1.33158 0.885512
1.25268 0.908388
1.27947 0.882604
1.25241 0.876734
1.26927 0.915657
1.28884 0.88941
1.21869 0.866007
1.28437 0.884098
1.28296 0.896861
1.31109 0.905855
1.29219 0.926
1.23675 0.880003
1.30739 0.915941
1.2876 0.87587
1.30799 0.907162
1.30787 0.900957
1.32889 0.891528
1.27516 0.868868
1.30612 0.89073
1.35053 0.907888
1.22153 0.884319
1.33086 0.917357
1.25139 0.87027
1.22702 0.930292
1.32561 0.885983
1.31254 0.894593
1.2653 0.877631
1.22026 0.879771
1.29784 0.907676
1.25376 0.895189
1.20714 0.888735
1.28772 0.915645
1.28619 0.890366
1.259 0.891858
1.28474 0.922514
1.25909 0.904846
1.26219 0.926151
1.32035 0.884415
1.32279 0.862578
1.28065 0.870504
1.2479 0.858193
1.34975 0.883737
1.21143 0.903801
1.29584 0.885905
1.27986 0.911445
1.27521 0.890818
1.28693 0.885505
1.31283 0.925045
1.24759 0.862306
1.312 0.927847
1.28998 0.9144
1.36809 0.887049
1.25219 0.905859
1.30386 0.887479
1.23131 0.900594
1.27164 0.902711
1.26537 0.874508
1.34274 0.875699
1.20408 0.878568
1.3324 0.880736
1.34016 0.910157
1.23586 0.88982
1.28405 0.904918
1.23743 0.891502
1.35555 0.873177
1.298 0.901068
1.23655 0.893607
1.27821 0.906993
1.24965 0.881674
1.26635 0.885439
1.24883 0.899231
1.24795 0.891216
1.28834 0.917066
1.31616 0.890811
1.25978 0.893421
1.19358 0.890076
1.28431 0.895919
1.22137 0.898382
1.27667 0.871932
1.31754 0.889699
1.27078 0.913125
1.34449 0.884558
1.31471 0.906929
1.27722 0.899061
1.29322 0.912265
1.29251 0.892235
1.3013 0.857401
1.30754 0.883708
1.321 0.920577
1.2749 0.909487
1.30773 0.89889
1.23442 0.907068
1.33218 0.889464
1.29985 0.897404
1.2953 0.906924
1.25735 0.877665
1.27992 0.901778
1.35627 0.923606
1.31298 0.887356
1.30567 0.907428
1.24799 0.886464
1.24527 0.89604
1.24649 0.867331
1.23434 0.866354
1.23262 0.906688
1.26906 0.864481
1.31386 0.880244
1.23446 0.875825
1.30225 0.871733
1.28758 0.942493
1.27359 0.8746
1.21847 0.914368
1.2833 0.876904
1.26146 0.869722
1.33898 0.910604
1.29383 0.893281
1.2845 0.88963
1.29536 0.875872
1.32227 0.892406
1.25826 0.920493
1.25819 0.906301
1.25176 0.888117
1.26221 0.904246
1.26325 0.892346
1.28616 0.872822
1.31126 0.8707
1.24719 0.897971
1.28051 0.914498
1.2817 0.879494
1.41589 0.886121
1.31908 0.909739
1.29879 0.886162
1.29675 0.896006
1.31087 0.875572
1.21402 0.907335
1.28895 0.904906
1.2114 0.895427
1.2151 0.902807
1.30628 0.880755
1.23686 0.912286
1.32852 0.892941
1.25243 0.904918
1.28739 0.89501
1.37896 0.883422
1.34163 0.900157
1.28176 0.91432
1.28685 0.929243
1.24621 0.903233
1.335 0.856527
1.28115 0.875209
1.27907 0.860913
1.3647 0.907754
1.27064 0.892142
1.30152 0.840292
1.25153 0.852177
1.29679 0.869337
1.27773 0.906446
1.27954 0.891281
1.23566 0.907888
1.33123 0.878992
1.29079 0.891348
1.39201 0.900257
1.29935 0.898852
1.29303 0.917898
1.24376 0.871551
1.36831 0.886407
1.25888 0.899488
1.2399 0.867973
1.30075 0.906447
1.26663 0.894795
1.28937 0.896779
1.28319 0.915181
1.2914 0.903844
1.22604 0.892134
1.28493 0.865643
1.25693 0.915223
1.17788 0.91207
1.31807 0.887731
1.31806 0.877444
1.33148 0.913566
1.31501 0.895104
1.21561 0.900399
1.32397 0.882778
1.2412 0.887779
1.28798 0.901722
1.32204 0.891469
1.30445 0.878949
1.3195 0.893537
1.29647 0.893651
1.30864 0.90272
1.33842 0.906772
1.27186 0.884606
1.25376 0.912435
1.24348 0.88483
1.32806 0.885722
1.26289 0.875498
1.34817 0.930449
1.2741 0.836675
1.31649 0.894658
1.28392 0.890792
1.3196 0.912111
1.32396 0.894139
1.26697 0.911493
1.23262 0.897811
1.22736 0.907085
1.22682 0.897704
1.23601 0.892503
1.21706 0.906444
1.2371 0.883
1.31262 0.876441
1.2599 0.89583
1.22319 0.91635
1.32081 0.877347
1.25163 0.909003
1.2904 0.878847
1.27256 0.882984
1.22443 0.863475
1.24356 0.915976
1.25261 0.886835
1.26352 0.875685
1.34715 0.879681
1.32883 0.882916
1.23169 0.914487
1.29897 0.870389
1.2282 0.893458
1.33507 0.866107
1.30039 0.866845
1.25275 0.871158
1.24227 0.896042
1.33085 0.8939
1.22338 0.893557
1.26838 0.905916
1.26894 0.895362
1.3019 0.881896
1.2906 0.887678
1.33256 0.882188
1.24752 0.90323
1.2359 0.907687
1.26341 0.883272
1.31385 0.899933
1.30353 0.852505
1.24364 0.898424
1.28896 0.898842
1.27565 0.90108
1.2543 0.879171
1.24601 0.927591
1.27888 0.894137
1.30922 0.911365
1.25757 0.857223
In [134]:
fig, ax = plt.subplots(figsize=(5, 5))

w_grid_ratio = ratio_estimator.predict(w_grid.reshape(300*300, 2))
w_grid_ratio = w_grid_ratio.reshape(300, 300)

ax.contourf(w1, w2, w_grid_ratio, cmap=plt.cm.magma)
ax.scatter(*sess.run(w_posterior_samples).T, alpha=.8)
ax.scatter(*sess.run(w_prior_samples).T, alpha=.8)

ax.set_xlabel('$w_1$')
ax.set_ylabel('$w_2$')

ax.set_xlim(w_min, w_max)
ax.set_ylim(w_min, w_max)

plt.show()
In [138]:
fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(9, 4))

sns.kdeplot(*sess.run(w_posterior_samples).T, cmap='magma', ax=ax2)

ax2.set_xlim(w_min, w_max)
ax2.set_ylim(w_min, w_max)

ax1.contourf(w1, w2, 
             np.exp(log_prior+np.sum(llhs, axis=2)), 
             cmap=plt.cm.magma)

ax1.scatter(*sess.run(w_posterior_samples).T, alpha=.6)

ax1.set_xlabel('$w_1$')
ax1.set_ylabel('$w_2$')

ax1.set_xlim(w_min, w_max)
ax1.set_ylim(w_min, w_max)

plt.show()
In [ ]:
In [ ]:

Visualizing the Latent Space of Vector Drawings from the Google QuickDraw Dataset with SketchRNN, PCA and t-SNE

Comments

t-SNE Visualization of Sheep Sketches

This is the third part in a series of notes on my exploration of the recently released Google QuickDraw dataset 1, using the concurrently released SketchRNN model.

The QuickDraw dataset is curated from the millions of drawings contributed by over 15 million people around the world who participated in the "Quick, Draw!" A.I. Experiment, in which they were given the challenge of drawing objects belonging to a particular class (such as "cat") in under 20 seconds.

SketchRNN is an impressive generative model that was trained to produce vector drawings using this dataset. It was of particular interest to me because it cleverly assembles many of the latest tools and techniques recently developed in machine learning, such as Variational Autoencoders, HyperLSTMs (a HyperNetwork for LSTM), Autoregressive models, Layer Normalization, Recurrent Dropout, the Adam optimizer, among others.

Read more…

Exploring the Google QuickDraw Dataset with SketchRNN (Part 3)

Comments

t-SNE Visualization of Sheep Sketches

This is the third part in a series of notes on my exploration of the recently released Google QuickDraw dataset 1, using the concurrently released SketchRNN model.

The QuickDraw dataset is curated from the millions of drawings contributed by over 15 million people around the world who participated in the "Quick, Draw!" A.I. Experiment, in which they were given the challenge of drawing objects belonging to a particular class (such as "cat") in under 20 seconds.

SketchRNN is an impressive generative model that was trained to produce vector drawings using this dataset. It was of particular interest to me because it cleverly assembles many of the latest tools and techniques recently developed in machine learning, such as Variational Autoencoders, HyperLSTMs (a HyperNetwork for LSTM), Autoregressive models, Layer Normalization, Recurrent Dropout, the Adam optimizer, among others.

Read more…

Exploring the Google QuickDraw Dataset with SketchRNN (Part 2)

Comments

This is the second part in a series of notes on my exploration of the recently released Google QuickDraw dataset, using the concurrently released SketchRNN model.

In the previous note, we set up our development environment, downloaded a subset of the data along with some pre-trained models, and developed some utilities for visualizing the data in the notebook. We retain most of the code from previous note and omit the expository code and markdown cells.

The QuickDraw dataset is curated from the millions of drawings contributed by over 15 million people around the world who participated in the "Quick, Draw!" A.I. Experiment, in which they were given the challenge of drawing objects belonging to a particular class (such as "cat") in under 20 seconds.

SketchRNN is a very impressive generative model that was trained to produce vector drawings using this dataset. It was of particular interest to me because it cleverly combines many of the latest tools and techniques recently developed in machine learning, such as Variational Autoencoders, HyperLSTMs (a HyperNetwork for LSTM), Autoregressive models, Layer Normalization, Recurrent Dropout, the Adam optimizer, and others.

In [48]:
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
%load_ext autoreload
%autoreload 2

The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload
In [49]:
import matplotlib.pyplot as plt
import matplotlib.patches as patches

import numpy as np
import tensorflow as tf

from matplotlib.animation import FuncAnimation
from matplotlib.path import Path
from matplotlib import rc

from six.moves import map
In [50]:
from magenta.models.sketch_rnn.sketch_rnn_train import \
    (load_env,
     load_checkpoint,
     reset_graph,
     download_pretrained_models,
     PRETRAINED_MODELS_URL)
from magenta.models.sketch_rnn.model import Model, sample
from magenta.models.sketch_rnn.utils import (lerp,
                                             slerp,
                                             get_bounds, 
                                             to_big_strokes,
                                             to_normal_strokes)
In [52]:
# For inine display of animation
# equivalent to rcParams['animation.html'] = 'html5'
rc('animation', html='html5')
In [53]:
# set numpy output to something sensible
np.set_printoptions(precision=8, 
                    edgeitems=6, 
                    linewidth=200, 
                    suppress=True)
In [54]:
tf.logging.info("TensorFlow Version: {}".format(tf.__version__))

INFO:tensorflow:TensorFlow Version: 1.1.0

Getting the Pre-Trained Models and Data

In [7]:
DATA_DIR = ('http://github.com/hardmaru/sketch-rnn-datasets/'
            'raw/master/aaron_sheep/')
MODELS_ROOT_DIR = '/tmp/sketch_rnn/models'
In [8]:
DATA_DIR
Out[8]:
'http://github.com/hardmaru/sketch-rnn-datasets/raw/master/aaron_sheep/'
In [9]:
PRETRAINED_MODELS_URL
Out[9]:
'http://download.magenta.tensorflow.org/models/sketch_rnn.zip'
In [10]:
download_pretrained_models(
    models_root_dir=MODELS_ROOT_DIR,
    pretrained_models_url=PRETRAINED_MODELS_URL)

INFO:tensorflow:/tmp/sketch_rnn/models/sketch_rnn.zip already exists, using cached copy
INFO:tensorflow:Unzipping /tmp/sketch_rnn/models/sketch_rnn.zip...
INFO:tensorflow:Unzipping complete.

We look at the layer normalized model trained on the aaron_sheep dataset for now.

In [11]:
MODEL_DIR = MODELS_ROOT_DIR + '/aaron_sheep/layer_norm'
In [12]:
(train_set, 
 valid_set, 
 test_set, 
 hps_model, 
 eval_hps_model, 
 sample_hps_model) = load_env(DATA_DIR, MODEL_DIR)

INFO:tensorflow:Downloading http://github.com/hardmaru/sketch-rnn-datasets/raw/master/aaron_sheep/aaron_sheep.npz
INFO:tensorflow:Loaded 7400/300/300 from aaron_sheep.npz
INFO:tensorflow:Dataset combined: 8000 (7400/300/300), avg len 125
INFO:tensorflow:model_params.max_seq_len 250.
total images <= max_seq_len is 7400
total images <= max_seq_len is 300
total images <= max_seq_len is 300
INFO:tensorflow:normalizing_scale_factor 18.5198.
In [222]:
class SketchPath(Path):
    
    def __init__(self, data, factor=.2, *args, **kwargs):
        
        vertices = np.cumsum(data[::, :-1], axis=0) / factor
        codes = np.roll(self.to_code(data[::,-1].astype(int)), 
                        shift=1)
        codes[0] = Path.MOVETO

        super(SketchPath, self).__init__(vertices, 
                                         codes, 
                                         *args, 
                                         **kwargs)
        
    @staticmethod
    def to_code(cmd):
        # if cmd == 0, the code is LINETO
        # if cmd == 1, the code is MOVETO (which is LINETO - 1)
        return Path.LINETO - cmd
In [69]:
def draw(sketch_data, factor=.2, pad=(10, 10), ax=None):

    if ax is None:
        ax = plt.gca()

    x_pad, y_pad = pad
    
    x_pad //= 2
    y_pad //= 2
        
    x_min, x_max, y_min, y_max = get_bounds(data=sketch_data,
                                            factor=factor)

    ax.set_xlim(x_min-x_pad, x_max+x_pad)
    ax.set_ylim(y_max+y_pad, y_min-y_pad)

    sketch = SketchPath(sketch_data)

    patch = patches.PathPatch(sketch, facecolor='none')
    ax.add_patch(patch)

The real fun begins

Everything up to here has more or less been copied straight from the previous notebook. Now we load the pre-trained SketchRNN model and use it to begin our exploration of the test dataset.

In [110]:
# construct the sketch-rnn model here:
reset_graph()
model = Model(hps_model)
eval_model = Model(eval_hps_model, reuse=True)
sample_model = Model(sample_hps_model, reuse=True)

INFO:tensorflow:Model using gpu.
INFO:tensorflow:Input dropout mode = False.
INFO:tensorflow:Output dropout mode = False.
INFO:tensorflow:Recurrent dropout mode = True.
INFO:tensorflow:Model using gpu.
INFO:tensorflow:Input dropout mode = False.
INFO:tensorflow:Output dropout mode = False.
INFO:tensorflow:Recurrent dropout mode = False.
INFO:tensorflow:Model using gpu.
INFO:tensorflow:Input dropout mode = False.
INFO:tensorflow:Output dropout mode = False.
INFO:tensorflow:Recurrent dropout mode = False.
In [111]:
sess = tf.InteractiveSession()
sess.run(tf.global_variables_initializer())
In [112]:
# loads the weights from checkpoint into our model
load_checkpoint(sess=sess, checkpoint_path=MODEL_DIR)

INFO:tensorflow:Loading model /tmp/sketch_rnn/models/aaron_sheep/layer_norm/vector.
INFO:tensorflow:Restoring parameters from /tmp/sketch_rnn/models/aaron_sheep/layer_norm/vector

The helper functions for encoding a sketch to some latent code $z$ and then decoding it back to a sketch were provided in the original notebook. I just made some minor syntactic changes and removed the behaviour of plotting as a side-effect.

In [73]:
def encode(input_strokes):
    strokes = to_big_strokes(input_strokes).tolist()
    strokes.insert(0, [0, 0, 1, 0, 0])
    seq_len = [len(input_strokes)]
    z = sess.run(eval_model.batch_z,
                 feed_dict={
                    eval_model.input_data: [strokes], 
                    eval_model.sequence_lengths: seq_len})[0]
    return z
In [74]:
def decode(z_input=None, temperature=.1, factor=.2):
    z = None
    if z_input is not None:
        z = [z_input]
    sample_strokes, m = sample(
        sess, 
        sample_model, 
        seq_len=eval_model.hps.max_seq_len, 
        temperature=temperature, z=z)
    return to_normal_strokes(sample_strokes)

Now we get a random sample from the test dataset

In [120]:
sketch = test_set.random_sample()
In [138]:
fig, ax = plt.subplots(figsize=(3, 3),
                       subplot_kw=dict(xticks=[], 
                                       yticks=[], 
                                       frame_on=False))

draw(sketch, ax=ax)

plt.show()

We project it into the 128-dimensional latent space using the pre-trained encoder

In [183]:
z = encode(sketch)
z.shape
Out[183]:
(128,)

Now we can reconstruct the original sketch from the learned latent representation using the pre-trained decoder, with temperature $\tau=0.8$. The temperature parameter controls the level of randomness in the samples generated by the model, which becomes deterministic as $\tau \to 0$, and produces samples that are the most likely point in the probability density function. See pg. 7 of the original paper for further discussion of the effects the temperature parameter has on the sampling process.

In [216]:
sketch_reconstructed = decode(z, temperature=.6)
sketch_reconstructed.shape
Out[216]:
(250, 3)
In [217]:
fig, ax = plt.subplots(figsize=(3, 3),
                       subplot_kw=dict(xticks=[], 
                                       yticks=[], 
                                       frame_on=False))

draw(sketch_reconstructed, ax=ax)

plt.show()

Variance in the Reconstruction

The grid of drawings below consists of samples of the reconstructed drawings at various settings of the temperature parameter. The first column is the original drawing, and each of the remaining columns are 5 samples of the reconstructed drawing with $\tau$ increasing from 0.1 to 0.9.

In [182]:
fig, ax_arr = plt.subplots(nrows=5, 
                           ncols=10, 
                           figsize=(8, 4),
                           subplot_kw=dict(xticks=[],
                                           yticks=[],
                                           frame_on=False))
fig.tight_layout()

for row_num, ax_row in enumerate(ax_arr):    
    for col_num, ax in enumerate(ax_row):
        if not col_num:
            draw(sketch, ax=ax)
            xlabel = 'original'
        else:
            t = col_num / 10.
            draw(decode(z, temperature=t), ax=ax)
            xlabel = r'$\tau={}$'.format(t)
        if row_num+1 == len(ax_arr):
            ax.set_xlabel(xlabel)

plt.show()

At the lowest setting of the temperature at $\tau=0.1$, we see the samples consistently share a similar appearance - they all look like vertical strokes emanating from a fluffy cloud. However, they are also consistently dissimilar to the original sketch. In this sense, the samples from the models seems to exhibit high bias and low variance. As we increase the variance in the samples by increasing $\tau$, we start to find some samples that resemble our original sketch. But when we increase $\tau$ a little too much, beyond say 0.8, we begin to see a little too much randomness in the samples.

Drawing Comparisons

Humans typically write and, by extension, draw from left to right, top to bottom. Here, I wanted to animate the process of the original sketch being drawn alongside the decoder's reconstruction of the sketch to compare stroke patterns, typical stroke lengths, etc.

In [218]:
fig, (ax1, ax2) = plt.subplots(ncols=2, nrows=1, figsize=(6, 3),
                               subplot_kw=dict(xticks=[], 
                                               yticks=[]))
fig.tight_layout()

x_pad, y_pad = 10, 10
    
x_pad //= 2
y_pad //= 2

(x_min_1, 
 x_max_1,
 y_min_1,
 y_max_1) = get_bounds(data=sketch, factor=.2)

(x_min_2, 
 x_max_2, 
 y_min_2, 
 y_max_2) = get_bounds(data=sketch_reconstructed, factor=.2)

x_min = np.minimum(x_min_1, x_min_2)
y_min = np.minimum(y_min_1, y_min_2)

x_max = np.maximum(x_max_1, x_max_2)
y_max = np.maximum(y_max_1, y_max_2)

ax1.set_xlim(x_min-x_pad, x_max+x_pad)
ax1.set_ylim(y_max+y_pad, y_min-y_pad)

ax1.set_xlabel('Original')

ax2.set_xlim(x_min-x_pad, x_max+x_pad)
ax2.set_ylim(y_max+y_pad, y_min-y_pad)

ax2.set_xlabel('Reconstruction')
Out[218]:
<matplotlib.text.Text at 0x7f232a2097d0>
In [219]:
def animate(i):

    original = SketchPath(sketch[:i+1])
    reconstructed = SketchPath(sketch_reconstructed[:i+1])

    patch1 = ax1.add_patch(patches.PathPatch(original,
                                             facecolor='none'))

    patch2 = ax2.add_patch(patches.PathPatch(reconstructed, 
                                             facecolor='none'))
    
    return patch1, patch2
In [220]:
frames = np.maximum(sketch.shape[0], 
                    sketch_reconstructed.shape[0])
frames
Out[220]:
249
In [221]:
FuncAnimation(fig,
              animate,
              frames=frames-1, 
              interval=15,
              repeat_delay=1000*3, 
              blit=True)
Out[221]:

Unfortunately, the strokes that make up a sketch have been normalized with the Ramer–Douglas–Peucker algorithm, which is a simple stroke simplification process. This means the strokes aren't quite the same as the that which the human originally used to construct the sketch. Moreover, the timing of each stroke are also important to understanding patterns in how humans draw quick sketches. While timestamp data is provided in the full QuickDraw dataset, they are not preserved in the modified version of the dataset used by SketchRNN.

Exploring the Google QuickDraw Dataset with SketchRNN (Part 1)

Comments

This is the first part in what will hopefully be a series of notes 1 on my exploration of the recently released Google QuickDraw dataset, using the concurrently released SketchRNN model.

The QuickDraw dataset is curated from the millions of drawings contributed by over 15 million people around the world who participated in the "Quick, Draw!" A.I. Experiment, in which they were given the challenge of drawing objects belonging to a particular class (such as "cat") in under 20 seconds.

SketchRNN is a very impressive generative model that was trained to produce vector drawings using this dataset. It was of particular interest to me because it cleverly combines many of the latest tools and techniques recently developed in machine learning, such as Variational Autoencoders, HyperLSTMs (a HyperNetwork for LSTM), Autoregressive models, Layer Normalization, Recurrent Dropout, the Adam optimizer, and others.

This notebook was based on the notebook included with the code release. I've made significant stylistic changes and some minor changes to ensure Python 3 compatibility as Magenta only supports Python 2 currently.

  1. These notes will likely be quite hasty and unpolished , as it has been written more for myself than anyone else. While I've always tried to avoid cluttering up my blog with notebooks on in-progress work, I've decided to just make a habit of posting them everytime I complete a session of work on something for the day. The aim is to start creating series of short and succinct notebooks, and avoid further accumulating a collection of long and disorganized notebooks that I never want to touch again, because the sheer effort of going through, cleaning up the experimental code, and articulating it effectively in a well-thought-out blog article just requires too much willpower. Doing it this way will hopefully make it much easier for me to create and share useful content quickly.↩

Environment Set-up

Some preamble for plotting (I really ought to put these in a config file at some point...), and importing dependencies. I've made the imports here explicit so we know exactly what methods/objects are imported and can find the module from which they were imported more easily.

In [1]:
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
%load_ext autoreload
%autoreload 2
In [2]:
import matplotlib.pyplot as plt
import matplotlib.patches as patches

import numpy as np
import tensorflow as tf

from matplotlib.path import Path
In [3]:
from magenta.models.sketch_rnn.sketch_rnn_train import \
    (load_env,
     load_checkpoint,
     reset_graph,
     download_pretrained_models,
     PRETRAINED_MODELS_URL)
from magenta.models.sketch_rnn.model import Model, sample
from magenta.models.sketch_rnn.utils import (get_bounds, 
                                             to_big_strokes,
                                             to_normal_strokes)
In [4]:
# set numpy output to something sensible
np.set_printoptions(precision=8, 
                    edgeitems=6, 
                    linewidth=200, 
                    suppress=True)
In [5]:
tf.logging.info("TensorFlow Version: {}".format(tf.__version__))

INFO:tensorflow:TensorFlow Version: 1.1.0

Getting the Pre-Trained Models and Data

In [6]:
DATA_DIR = ('http://github.com/hardmaru/sketch-rnn-datasets/'
            'raw/master/aaron_sheep/')
MODELS_ROOT_DIR = '/tmp/sketch_rnn/models'
In [7]:
DATA_DIR
Out[7]:
'http://github.com/hardmaru/sketch-rnn-datasets/raw/master/aaron_sheep/'
In [8]:
PRETRAINED_MODELS_URL
Out[8]:
'http://download.magenta.tensorflow.org/models/sketch_rnn.zip'
In [9]:
download_pretrained_models(
    models_root_dir=MODELS_ROOT_DIR,
    pretrained_models_url=PRETRAINED_MODELS_URL)

INFO:tensorflow:/tmp/sketch_rnn/models/sketch_rnn.zip already exists, using cached copy
INFO:tensorflow:Unzipping /tmp/sketch_rnn/models/sketch_rnn.zip...
INFO:tensorflow:Unzipping complete.

The directory tree looks like this. There are a few pretrained models for us to explore.

In [10]:
!tree -L 3 /tmp/sketch_rnn/

/tmp/sketch_rnn/
└── models
    ├── aaron_sheep
    │   ├── layer_norm
    │   ├── layer_norm_uncond
    │   ├── lstm
    │   └── lstm_uncond
    ├── catbus
    │   └── lstm
    ├── elephantpig
    │   └── lstm
    ├── flamingo
    │   └── lstm_uncond
    ├── owl
    │   └── lstm
    └── sketch_rnn.zip

14 directories, 1 file

We look at the layer normalized model trained on the aaron_sheep dataset for now.

In [11]:
MODEL_DIR = MODELS_ROOT_DIR + '/aaron_sheep/layer_norm'
In [12]:
(train_set, 
 valid_set, 
 test_set, 
 hps_model, 
 eval_hps_model, 
 sample_hps_model) = load_env(DATA_DIR, MODEL_DIR)

INFO:tensorflow:Downloading http://github.com/hardmaru/sketch-rnn-datasets/raw/master/aaron_sheep/aaron_sheep.npz
INFO:tensorflow:Loaded 7400/300/300 from aaron_sheep.npz
INFO:tensorflow:Dataset combined: 8000 (7400/300/300), avg len 125
INFO:tensorflow:model_params.max_seq_len 250.
total images <= max_seq_len is 7400
total images <= max_seq_len is 300
total images <= max_seq_len is 300
INFO:tensorflow:normalizing_scale_factor 18.5198.

Drawing the Dataset

The strokes object variable is the list of data points, which are sequences of strokes, represented as a 2D NumPy array of xy-offsets and the pen state.

In [13]:
len(train_set.strokes)
Out[13]:
7400

We can get a random sample from the dataset like so

In [122]:
a = train_set.random_sample()
a.shape
Out[122]:
(122, 3)

In the original notebook, the authors implemented their own function to iterate through a stroke sequence and write out a SVG Path string. I found this a bit inelegant and cumbersome to work with.

Here we simply subclass Path, which underpins all matplotlib.patch classes. This is almost perfect for our data format since it "supports the standard set of moveto, lineto, curveto commands to draw simple and compound outlines consisting of line segments and splines. The Path is instantiated with a (N,2) array of (x,y) vertices, and a N-length array of path codes." We just need to normalize the data slightly as we shall explain later.

In [133]:
class StrokesPath(Path):
    
    def __init__(self, data, factor=.2, *args, **kwargs):
        
        vertices = np.cumsum(data[::, :-1], axis=0) / factor
        codes = np.roll(self.to_code(data[::,-1].astype(int)), 
                        shift=1)

        super(StrokesPath, self).__init__(vertices, 
                                          codes, 
                                          *args, 
                                          **kwargs)
        
    @staticmethod
    def to_code(cmd):
        # if cmd == 0, the code is LINETO
        # if cmd == 1, the code is MOVETO (which is LINETO - 1)
        return Path.LINETO - cmd

Now drawing the strokes becomes as simple as

In [134]:
fig, ax = plt.subplots(figsize=(3, 3))

strokes = StrokesPath(a)

patch = patches.PathPatch(strokes, facecolor='none')
ax.add_patch(patch)

x_min, x_max, y_min, y_max = get_bounds(data=a, factor=.2)

ax.set_xlim(x_min-5, x_max+5)
ax.set_ylim(y_max+5, y_min-5)

ax.axis('off')

plt.show()

We define this as a function to maximize modularity and reusability

In [135]:
def draw(stroke, factor=.2, pad=(10, 10), ax=None):

    if ax is None:
        ax = plt.gca()

    x_pad, y_pad = pad
    
    x_pad //= 2
    y_pad //= 2
        
    x_min, x_max, y_min, y_max = get_bounds(data=stroke,
                                            factor=factor)

    ax.set_xlim(x_min-x_pad, x_max+x_pad)
    ax.set_ylim(y_max+y_pad, y_min-y_pad)

    strokes = StrokesPath(stroke)

    patch = patches.PathPatch(strokes, facecolor='none')
    ax.add_patch(patch)
    
    ax.axis('off')

Now it is easy to fully take advantage of the functionality provided by Matplotlib to create more complex plots. For example, to draw the sketches in a grid, we just call our draw function on the grid of axes created with subplots.

In [136]:
fig, ax_arr = plt.subplots(nrows=5, 
                           ncols=10, 
                           figsize=(8, 4),
                           subplot_kw=dict(xticks=[],
                                           yticks=[],
                                           frame_on=False))
fig.tight_layout()

for ax_row in ax_arr:
    for ax in ax_row:
        strokes = train_set.random_sample()
        draw(strokes, ax=ax)

plt.show()
Explanation

The last column of the 2D array is essentially a binary value that gives the pen action taken prior to the next stroke in the sequence.

In [137]:
a[::,-1].astype(int)
Out[137]:
array([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1])

A 1 means the pen is to be lifted from the current point before being moved to the next point. In terms of SVG Path commands or Matplotlib Path codes, this corresponds to the next point being preceded by the m command or having the MOVETO code, respectively. Otherwise, the pen just draws a line to the next point, which means it is preceded by the l command or has the LINETO code.

Our to_code static method above simply turns the pen action for the next state to the corresponding Matplotlib Path code.

In [138]:
{c: getattr(Path, c) for c in dir(Path) if c.isupper()}
Out[138]:
{'CLOSEPOLY': 79,
 'CURVE3': 3,
 'CURVE4': 4,
 'LINETO': 2,
 'MOVETO': 1,
 'NUM_VERTICES_FOR_CODE': {0: 1, 1: 1, 2: 1, 3: 2, 4: 3, 79: 1},
 'STOP': 0}
In [139]:
StrokesPath.to_code(a[::,-1].astype(int))
Out[139]:
array([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1])

As explained, the $i$th code is meant for the $i+1$th vertex, so we are not done just yet. We must right shift the array by 1. But what do we do with the first and last elements of the sequence?

Note that the pen action for all stroke sequences are terminated with the pen being lifted. This corresponds to the next vertex having a MOVETO code. However, there are no vertices left, so this is superfluous and may be discarded.

In [140]:
all(StrokesPath.to_code(a[-1,-1]) == Path.MOVETO \
    for a in train_set.strokes)
Out[140]:
True

On the other hand, the first vertex is always required to have a MOVETO code (otherwise, where the drawing actually begins is not well-defined). Therefore, the most simple and elegant solution is just to np.roll the array to right by 1, so the first code takes on the value of the last code, which is always a MOVETO.

Lastly, while SVG Paths supports both absolute positions and relative offsets (with M and m commands respectively), Matplotlib only supports absolute positions. As the datasets gives the points as relative offsets, we convert it to absolute positions simply by calling taking the cumulative sum with np.cumsum.

Trigonometric functions with recursion and higher-order functions in Python

Comments

In [1]:
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
In [2]:
import matplotlib.pyplot as plt
import numpy as np

from itertools import count, islice, takewhile
from functools import reduce, partial

The Taylor series expansion for the trigonometric function $\sin{x}$ around the point $a=0$ (also known as the Maclaurin series in this case) is given by:

$$ \sin{x} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dotsb \text{ for all } x $$

The $k$th term of the expansion is given by

$$ \frac{(-1)^k}{(2k+1)!} x^{2k+1} $$

It is easy to evaluate this closed-form expression directly. However, it is more elegant and indeed more efficient to compute the terms bottom-up, by iteratively calculating the next term using the value of the previous term. This is just like computing factorials or a sequence of Fibonacci numbers using the bottom-up approach in dynamic programming.

Read more…

Matplotlib Unchained

Comments

I a previous post, I outlined how to embed a Matplotlib Animation directly in the Jupyter Notebook as a HTML5 video. In this notebook, we take the same Animation and save it as a GIF using Imagemagick. First, let us reproduce the FuncAnimation object from the notebook.

In [1]:
%matplotlib inline
In [2]:
import numpy as np
import matplotlib.pyplot as plt

from matplotlib import animation, rc
from IPython.display import HTML, Image
In [3]:
# equivalent to rcParams['animation.html'] = 'html5'
rc('animation', html='html5')
In [4]:
# Create new Figure with black background
fig = plt.figure(figsize=(8, 8), facecolor='black')

# Add a subplot with no frame
ax = fig.add_subplot(111, frameon=False)

# Generate random data
data = np.random.uniform(0, 1, (64, 75))
X = np.linspace(-1, 1, data.shape[-1])
G = 1.5 * np.exp(-4 * X * X)

# Set y limit (or first line is cropped because of thickness)
ax.set_ylim(-1, 70)

# No ticks
ax.set_xticks([])
ax.set_yticks([])

# 2 part titles to get different font weights
ax.text(0.5, 1.0, "MATPLOTLIB ", transform=ax.transAxes,
        ha="right", va="bottom", color="w",
        family="sans-serif", fontweight="light", fontsize=16)
ax.text(0.5, 1.0, "UNCHAINED", transform=ax.transAxes,
        ha="left", va="bottom", color="w",
        family="sans-serif", fontweight="bold", fontsize=16)

# Generate line plots
lines = [ax.plot((1-i/200.)*X, i+G*d , color="w", lw=1.5-i/100.)[0] 
         for i, d in enumerate(data)]
In [5]:
def animate(*args):
    # Shift all data to the right
    data[:, 1:] = data[:, :-1]

    # Fill-in new values
    data[:, 0] = np.random.uniform(0, 1, len(data))

    # Update data
    for i, line in enumerate(lines):
        line.set_ydata(i + G * data[i])

    # Return modified artists
    return lines
In [6]:
# call the animator. blit=True means only re-draw the parts that have changed.
anim = animation.FuncAnimation(fig, animate, interval=20, blit=True)

Now, we just need to save the animation instance with writer=imagemagick. But before we do that, we first make sure imagemagick has been properly installed on our system.

In [7]:
!brew install imagemagick 

Updating Homebrew...
Warning: imagemagick-7.0.5-3 already installed

Now we can go ahead and save it as a GIF.

In [8]:
anim.save('../../files/unchained.gif', writer='imagemagick', fps=60, savefig_kwargs=dict(facecolor='black'))

Let's read it back in and display it to make sure it saved as expected.

In [9]:
Image(url='../../../unchained.gif')
Out[9]:

Save Matplotlib Animations as GIFs

Comments

I a previous post, I outlined how to embed a Matplotlib Animation directly in the Jupyter Notebook as a HTML5 video. In this notebook, we take the same Animation and save it as a GIF using Imagemagick. First, let us reproduce the FuncAnimation object from the notebook.

In [1]:
%matplotlib inline
In [2]:
import numpy as np
import matplotlib.pyplot as plt

from matplotlib import animation, rc
from IPython.display import HTML, Image
In [3]:
# equivalent to rcParams['animation.html'] = 'html5'
rc('animation', html='html5')
In [4]:
# First set up the figure, the axis, and the plot element we want to animate
fig, ax = plt.subplots()

ax.set_xlim(( 0, 2))
ax.set_ylim((-2, 2))

line, = ax.plot([], [], lw=2)
In [5]:
# initialization function: plot the background of each frame
def init():
    line.set_data([], [])
    return (line,)
In [6]:
# animation function. This is called sequentially
def animate(i):
    x = np.linspace(0, 2, 1000)
    y = np.sin(2 * np.pi * (x - 0.01 * i))
    line.set_data(x, y)
    return (line,)
In [7]:
# call the animator. blit=True means only re-draw the parts that 
# have changed.
anim = animation.FuncAnimation(fig, animate, init_func=init,
                               frames=100, interval=20, blit=True)
In [8]:
anim
Out[8]:

Now, we just need to save the animation instance with writer=imagemagick. But before we do that, we first make sure imagemagick has been properly installed on our system.

In [9]:
!brew install imagemagick

Warning: imagemagick-7.0.4-6 already installed

Now we can go ahead and save it as a GIF.

In [10]:
anim.save('../../files/animation.gif', writer='imagemagick', fps=60)

Finally, let's read it back in and display it to make sure it saved as expected.

In [11]:
Image(url='../../../animation.gif')
Out[11]: